I have lived and worked in both for several years each but am from a completely different part of Scotland so I have no real bias towards either. The first thing you have done right is pick the two best cities in Scotland. Both very different though.
My gut instinct before living in either told me I would prefer Edinburgh and that is where I lived first. Before living in either, I believed Edinburgh would be a little more upper class, less trouble and still have good travel links to Glasgow. I thought I would fit in more with the people there.
I will break it down from your key points and then add some of my own...
1) Music and Arts - both are great but Glasgow will always edge it on music. It's the biggest city in Scotland and attracts every big act along with every indie act and everything in-between. In terms of art, I'm not a huge art person but Edinburgh has the National galleries and the Edinburgh Festival. I have to say, the National galleries is an alright day out but the Edinburgh Festival is NOT a good time to live in Edinburgh. You cant move, the place is stuffed full of tourists and lots of drunk people. Glasgow is famous for museums, art scene and has Kelvingrove Art Gallery, Gallery of Modern Art and a bunch of others. Edinburgh has it's castle.... but there are castles everywhere in Scotland so you will get bored of visiting these after the first few and it's more of a one off day out than a reason to actually base yourself somewhere. I'd say your initial impression of giving the edge to Edinburgh is incorrect, I'd actually give this to Glasgow.
2) Crime - if you live in a nice part of either city and stay out of Hearts/Hibs/Celtic/Rangers scheme type areas then you will have no trouble. Just apply common sense. It's not Hong Kong, don't go wandering around in the late hours alone in a poor area (which I imagine you wouldn't be near anyway). Do you have an accommodation budget in mind? If you have a high budget, the West End of Glasgow or a nice part of Haymarket, Leith or New Town in Edinburgh City Centre may be good options for you. There are also really nice parts of Glasgow like Shawlands, Strathbungo etc. on the South side which are a bit cheaper but still really nice and trendy with easy links to the city. I would stick to areas close to the main city if you wont drive. There are really nice leafy suburbs around Glasgow but you really need to drive to live in them.
3) Public transport - coin flip, neither are great.
After living in both, I found that against my initial thoughts, I preferred Glasgow. Both locations have an abundance of trendy cafe's, excellent food, things to do and good travel links. However, the people are what make a city and I found myself gravitating more towards the people of Glasgow. They seemed friendlier to me, more down to earth and much more grounded. I found a lot of people in Edinburgh to be a bit "uppity" and the city centre felt much more touristy to me than Glasgow. When I lived in Glasgow, I still frequently travelled through to Edinburgh for a day out but I found myself happier to be based in Glasgow and exploring the hidden gems dotted throughout the city. It was just a bigger place with more things to explore. The cost of living was definitely higher in Edinburgh too without really feeling like my lifestyle was higher than in Glasgow.
Happy to answer any questions on both and any other questions you may have on living in Scotland. I would love to hear how you get on after you make the big move too. Scotland is a great country and I will without doubt move back there at some point.
An update having read through the thread properly. Some clearly tourists giving opinions on first impressions. Edinburgh is a good city and it may appear very welcoming on first impressions but when you look deeper you realise this is sometimes because the city is used to tourists and exchanging pleasantries.
Glasgow is a great city with character. In terms of trouble, you will find trouble in either city if you go for a walk through Easterhouse or Niddrie at the wrong time (or sometimes, anytime). Even in the city centre of Edinburgh, any local knows that walking through the Meadows late at night could result in either a mugging or a sexual assault if you have bad luck. I managed to live in both for many years with zero issues just by applying common sense.